MATH 311-504Topics in Applied MathematicsLecture 9:Inverse matrix.Identity matrixDefinition. The identity matrix (or unit matrix) isa diagonal matrix with all diagonal entries equal to 1.The n×n identity matrix is denoted Inor simply I .I1= (1), I2=1 00 1, I3=1 0 00 1 00 0 1.In general, I =1 0 . . . 00 1 . . . 0............0 0 . . . 1.Theorem. Let A be an arbitrary m ×n matrix.Then ImA = AIn= A.Inverse matrixNotation. Mn(R) denote the set of all n×nmatrices with real entries.Definition. Let A ∈ Mn(R). Suppose there existsan n×n matrix B such thatAB = BA = In.Then the matrix A is called invertible and B iscalled the inverse of A (denoted A−1).AA−1= A−1A = IExamplesA =1 10 1, B =1 −10 1, C =−1 00 1.AB =1 10 11 −10 1=1 00 1,BA =1 −10 11 10 1=1 00 1,C2=−1 00 1−1 00 1=1 00 1.Thus A−1= B, B−1= A, and C−1= C .Example. A =2 11 1.In the previous lecture it was shown that A2− 3A + I = O.Assume that the matrix A is invertible. ThenA2− 3A + I = O =⇒ A−1(A2− 3A + I ) = A−1O=⇒ A−1AA − 3A−1A + A−1I = O=⇒ A − 3I + A−1= O =⇒ A−1= 3I − AThe above argument suggests (but does not prove) that thematrix B = 3I − A =1 −1−1 2is the inverse of A.And, indeed, AB = BA = (3I − A)A = 3A − A2= I .Basic properties of inverse matrices:• If B = A−1then A = B−1. In other words, if Ais invertible, so is A−1, and A = (A−1)−1.• The inverse matrix (if it exists) is unique.Moreover, if AB = CA = I for some matricesB, C ∈ Mn(R) then B = C = A−1.Indeed, B = IB = (CA)B = C(AB) = CI = C .• If matrices A, B ∈ Mn(R) are invertible, so isAB, and (AB)−1= B−1A−1.(B−1A−1)(AB) = B−1(A−1A)B = B−1IB = B−1B = I ,(AB)(B−1A−1) = A(BB−1)A−1= AIA−1= AA−1= I .• Similarly, (A1A2. . . Ak)−1= A−1k. . . A−12A−11.Other examplesD =0 10 0, E =1 −1−1 1.D2=0 10 00 10 0=0 00 0.It follows that D is not invertible as otherwiseD2= O =⇒ D−1D2= D−1O =⇒ D = O.E2=1 −1−1 11 −1−1 1=2 −2−2 2= 2E .It follows that E is not invertible as otherwiseE2= 2E =⇒ E2E−1= 2EE−1=⇒ E = 2I .Theorem Suppose that D and E are n -by-nmatrices such that DE = O. Then exactly one ofthe following is true:(i) D is invertible, E = O;(ii) D = O, E is invertible;(iii) neither D nor E is invertible.Proof: If D is invertible thenDE = O =⇒ D−1DE = D−1O =⇒ E = O.If E is invertible thenDE = O =⇒ DEE−1= OE−1=⇒ D = O.It remains to notice that the zero matrix is notinvertible.Inverting diagonal matricesTheorem A diagonal matrix D = diag(d1, . . . , dn)is invertible if and only if all diagonal entries arenonzero: di6= 0 for 1 ≤ i ≤ n.If D is invertible then D−1= diag(d−11, . . . , d−1n).d10 . . . 00 d2. . . 0............0 0 . . . dn−1=d−110 . . . 00 d−12. . . 0............0 0 . . . d−1nInverting diagonal matricesTheorem A diagonal matrix D = diag(d1, . . . , dn)is invertible if and only if all diagonal entries arenonzero: di6= 0 for 1 ≤ i ≤ n.If D is invertible then D−1= diag(d−11, . . . , d−1n).Proof: If all di6= 0 then, clearly,diag(d1, . . . , dn) diag(d−11, . . . , d−1n) = diag(1, . . . , 1) = I ,diag(d−11, . . . , d−1n) diag(d1, . . . , dn) = diag(1, . . . , 1) = I .Now suppose that di= 0 for some i. Then for anyn×n matrix B the ith row of the matrix DB is azero row. Hence DB 6= I .Inverting 2-by-2 matricesDefinition. The determinant of a 2×2 matrixA =a bc dis det A = ad − bc.Theorem A matrix A =a bc dis invertible ifand only if det A 6= 0.If det A 6= 0 thena bc d−1=1ad − bcd −b−c a.Theorem A matrix A =a bc dis invertible ifand only if det A 6= 0. If det A 6= 0 thena bc d−1=1ad − bcd −b−c a.Proof: Let B =d −b−c a. ThenAB = BA =ad−bc 00 ad−bc= (ad − bc)I2.In the case det A 6= 0, we have A−1= (det A)−1B.In the case det A = 0, the matrices A and B arenot invertible because A = O ⇐⇒ B = O.Fundamental results on inverse matricesTheorem 1 Given a square matrix A, the following areequivalent:(i) A is invertible;(ii) x = 0 is the only solution of the matrix equation Ax = 0;(iii) the row echelon form of A has no zero rows;(iv) the reduced row echelon form of A is the identity matrix.Theorem 2 Suppose that a sequence of elementary rowoperations converts a matrix A into the identity matrix.Then the same sequence of operations converts the identitymatrix into the inverse matrix A−1.Theorem 3 For any n×n matrices A and B,BA = I ⇐⇒ AB = I .Row echelon form of a square matrix:noninvertible case invertible
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